52-8
Philosophie Naturalis
De Mot»Corpo-
E.U M.
m
m
m >
A + Oj n resolvo (f) in seriem infinitam A n + »OA n
m m — m m
m — z n
z n n
O O A n atque hujus termino in quo
0
en 54*; Resolvo in seriem infinitam &c. dignitatem evehendo, si pars una polyno-Ut h*c liqueant sequentia de dignitatum inii litterae a inomii ponatur aequalis,formulis sunt inemori* revocanda. exter* veri, partes omnes supponantur *-
Lemma. Binomii a-{-b, dignitas ~a quales littet* b. Exempli causa. Sit tri-. , . , n , nomi.im ad tertiam dignitatem
jus index », elli" + - a" 1 b ' elevandum, pone ,1 — z , d — a , e -\-s—b,
cujus
« X » - 1 _ , ■ ny.n-i%n-t. & formula — a 1 b 1 ■+■ —
IX» 1x1x3 , 1 1
» X H—T
X 2
n X n—i X n—i X n— ja" 4 & 4 a « — 2 b 2 11 * ” 1 »-Jji,
1 X 1 X t
for- mutabitur in seriem d 3 + 3 d 2 ( e -\~f)
«“3 ^,3 - 4 .
~ I X a X 3 X 4
+ Scc. Satis patet ex potentiarum ... , , , _
matione. Si enim binomium a+b, ad + 3 ^ ( e +/) 2 + CjL+J ) > 9 uinia», 2*m. &c. dignitates evehatur > perventum eit ad coerhcientem in ejea ett
in singulis dignitatis cujusque terminis, «—?> abrumpitur series ob n—5=0. Por-index littet* a unitate perpetuö decres- rb per eandem formulam generalem 0'+/) 2cit, dum contra index littet* b unitate — e e H” 1 f/H“//? ^ ^ "h/) 3 — e + 1J/crescit, & coefficientes seu unci* singulo- ~\~S e s 2 fi~f 3 - Quare tandemrum terminorum progrediuntur ut numeri — ^ 3 -h* d 2e "h J , s 1^” *
+ 3 d //+« 3 + 3 e 2 /+ l«/ 2 +/ J -
Ita etiam formulam pro dignitate insi-nitinomii possumus obtinere , sit enim se-ries A+BZ + C Z 2 +DZ 3 -t-KZ 4 &c.
rum terminorum progrediuntun nHn— 1 » X n — 1 X n —
I ’ I X » 1X2X3
lOtu—IXS—iX«-?
I X 2 X 3 X 4 ,
549. Cor. i. Si ponatur a — P, & Q ad dignitatem p evehenda sub ducto cal-'• h* u i cuJo * nvenietur-
Av+pAP-'BZ+pAv-'CZ* +p. 4 p-'DZ 3
4 -p X A P~ 2 B 2 Z 2 +p X^ A P -
b , , b 2 b 3
—-, adeoque s" — P" , - —Q 2 , —
a a 2 4 3
& 4
— O 3 > -z 04, his valonbus in lcm-
"— ,j 4 —
matis formula substitutis erit n+’inrp »
, _fi_ p . P 4- fnOl-l.'? X "- 3 X"- 1
' I ' 1X2 ^
+fx—'x-
i 4
pn£3 + &c. & si rursus ponatur P**Ja; fij . loco ».
. n p n O — B; ” X ”—- p n o 2 — C •
i ^ 1 X 2 ^ '
n X » — I X»—2
;;i. Cor. 3. Si ex binomio a + fc, ex-
m
CUJUS 1'
1X2X3
pnp 3 —D, & itä por-
ri», erit a + ^ *> = P =p»> -|_ — AQ_
P
in formula generali scribatur-, & erit
m P
__ _ m m m — p
a-\-b P -a ~'+■— a —p— h ' +m X m — p m — 2 P tnxm—pxm—zp1X2 xp 2 " ""p 4 - lXl xTp-j-
“ — ' 3 P . . mXnr — P xm—2: X m— 3 p
a-- L 3 + -»- 1 -if
xX»X3X4p 4 m
&C. >>
* jro. Cor. 2 . Iisdem formulis uti pos- ^ — 4 p , . . „ , . . — 1 —,
fumus pro polynomio quovis ad datam a t + &c. vel etiam ern a-\-b P
= S